ASTM003 AM and Accretion Lecture 10 of 11 (QMUL)

7 7.1

Week 10 The growth and settling of dust grains

The basic idea of how planetary formation occurs involves the build-up of large, rocky planetary cores/embryos by coagulation of the solid material in the protoplanetary disc. Initially, this involves the sticking together of dust grains, that gradually begin to settle towards the midplane of the disc as they grow. Dust then coagulates to form planetesimals, which in turn grow through mutual accretion into planetary embryos whose primary mode of growth is accretion from the planetesimal swarm.

We shall now consider the settling of dust grains towards the central plane of the protoplanetary disc.

Upper disc surface

z=H 2

g= Ω z

Disc midplane

z=0

We can estimate the settling time in a quiescent (non-turbulent) disc. The component of the gravitational force in the z direction acting on a dust grain of mass mgr is GM mgr z r3 where M is the mass of the central star and r is the distance of the grain from the star. We may approximate r ≃ R and use the Keplerian result that GM/R3 = Ω2 , where Ω is the angular velocity at the gas about the central star. The acceleration towards the midplane of the disc due to gravity is therefore gz = − Ω2 z .

The dust will therefore fall towards the central plane when we view the system in a rotating frame. As the dust grains move through the gas, the gas will exert a drag force on the grains, resisting their fall to the central plane. As a result, the grains will have a maximum velocity – their terminal velocity.

The drag force Fdrag acting on a spherical particle smaller than the mean free path of gas molecules moving with a speed much smaller than the sound speed is Fdrag = πa2 ρ cs v where a is the radius of the particle, ρ is the density of the gas, cs is the sound speed of the gas and v is the velocity of the grain relative to the gas. The equation of motion for a grain of mass mgr then becomes mgr

dv = Fdrag + mgr gz . dt 46

If the material of the grain has an internal density ρgr , this becomes 4π 3 4πa3 ρgr dv = πa2 ρ cs v − a ρgr Ω2 z , 3 dt 3

(144)

where ρ is the gas density.

In practice, a grain will quickly reach the terminal velocity, so it will fall to the central plane with the terminal velocity for most of its journey through the disc. From equation (144), using the result that dv/dt = 0 when the terminal velocity is reached, we find that the terminal velocity is 4aρgr Ω2 H 4aρgr Ω 8aρgr ΩH v = ∼ ∼ 3ρcs 3ρ 3Σ

(145)

where we have set z = H, used cs ≃ HΩ from equation (60), and have approximated Σ = 2Hρ. The settling time to the central plane is τs ∼ H/v ∼

3Σ Σ ∼ orbits .

8aρgr Ω 16aρgr

(146)

For Σ ∼ 102 g cm−2 = 103 kg m−2 , ρgr ∼ 1 g cm−3 = 103 kg m−3 , a = 10 µm = 10−5 m we get τs ∼ 104 orbits = 105 yr at 5 AU. As the grains settle towards the midplane of the disc, they collide with other grains, stick together, and grow.

7.2

Growth of settling grains

Consider a grain of radius a initially suspended at a height H above the disc midplane.

After falling a distance ∆z, it will increase its mass through collisions with other grains by an amount ∆m = πa2 nmgr ∆z, where n is the number density of grains of mass mgr . Thus we can write an expression that gives the rate of change of the mass of the grain as a function of its height above the midplane dm = −πa2 nmgr .

(147) dz The negative sign on the right hand side of this equation arises because the grain mass increases as its height above the midplane decreases. We note that the grain radius a=

3m 4πρgr

1/3

,

where ρgr is the internal density of the grains, and equation (147) becomes dm = −π dz

3m 4πρgr 47

2/3

nmgr .

(148)

The quantity nmgr is the mass volume density of solid material in the disc, and we may approximate this as

1 Σ nmgr = , ηgd 2H where ηgd = 100 is the gas-to-dust ratio (this is normally taken to be approximately 100 based on measurements of the dust content of interstellar gas clouds).

To obtain the mass of a typical grain when it reaches the midplane, we integrate equation (148) from height z = H to z = 0: 2/3

Z z=0 Z m(0) Σ 3 −2/3 dz. m dm = −π 4πρgr 2ηgd H z=H m(H) Assuming that when a grain reaches the midplane its mass is much larger than its initial mass at z = H (i.e. m(0) ≫ m(H)), we obtain 2 3

Σ 3 mz=0 ≃ .

4πρgr 2ηgd The typical grain size in the dust layer that forms at the disc midplane through grain settling is thus 1/3

3mz=0 az=0 = .

4πρgr Adopting values that approximately apply at 1 AU in a MMSN disc model (Σ = 2×104 kg m−2 , ρgr = 103 kg m−3 ), we obtain mz=0 = 0.057 kg and az=0 = 0.0247 m. Thus we see that the dust layer will be composed largely of centimetre sized pebbles !

7.3

Planetesimal formation

Further collisions and coagulation between the solid material within this pebble layer causes continued growth, and estimates based on computer simulations of particle > 1 − 10 km may be formed on a time sticking suggest that large objects with sizes of a ∼ scale similar to the grain settling time. Such objects are called planetesimals. Once such large objects have been formed, the effect of gravitational focusing becomes important when calculating the rate at which such objects collide to form larger objects.

7.4

Growth of planetesimals into planetary embryos

We will consider the growth of a large planetary embryo by accretion from a background of smaller objects. Let the embryo mass be mc , the embryo radius be Rc , the background object/planetesimal mass be m, and the number density of background objects be n. The accretion rate is

dmc 2Gmc 2 = n m v π Rc 1 + , (149) dt Rc v 2 where v is the velocity dispersion of the background objects. The term in brackets 1 + 2Gmc /(Rc v 2 ) on the r.h.s. of equation (149) is the gravitational focusing factor.

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v

L Rc

L

v

Consider the situation drawn in the diagram overleaf. An incoming object has velocity v and impact parameter L. The condition for the object being accreted is that it just grazes the accreting embryo, mc , so that its distance of closest approach must be Rc . Conservation of energy requires the velocity of the object at closest approach to be r 2Gmc v2 + Rc and conservation of angular momentum requires that r 2Gmc L v = Rc v 2 + Rc so that 2

L =

Rc2

2Gmc .

1+ Rc v 2

L is now the effective radius for collisions, and leads to the expression (149) for the mass accretion rate by the accreting embryo.

Let us now consider two different growth regimes, the so-called ‘orderly’ growth and ‘runaway’ growth regimes.

First, let us suppose that 2Gmc /Rc ≃ v 2 , such that the growing planetary embryo is responsible for stirring up the local planetesimal swarm (which it will do when it becomes massive compared to the background planetesimals), then we can write: dmc = 2nmπvRc2 . dt

(150)

Writing the embryo radius in terms of its mass and density 2/3

3mc 2 Rc = 4πρc we obtain

2/3

dmc 3 = 2nmπv m2/3 c . dt 4πρc We can write the mass-doubling time scale as −1

dmc τdouble = mc dt 49

(151)

1/3

and we obtain τdouble ∝ mc such that lower mass embryos will double their masses on shorter time scales than larger embryos.

If we work in terms of the planetary embryo radius, Rc , then we can write 4πRc2 ρc

dRc = 2nmπvRc2 dt

(152)

which simplifies to dRc = dt

πnmv 2ρc

= constant.

(153)

Thus we have that Rc ∝ t and the planetesimal radius grows linearly with time.

This is referred to as ‘orderly growth’, and occurs when the velocity dispersion of the background planetesimals is high or comparable to the escape velocity from the accreting embryo.

Let us now suppose that 2Gmc /Rc ≫ v 2 , so that the gravitational focusing factor becomes important, then equation (149) may be approximated by πRc Gmc dmc = 2nmv . dt v2

(154)

If we write the embryo radius in terms of its mass and density then we get

1/3 dmc 3 2πGnm = m4/3 c dt v 4πρc and we can see that the mass-doubling time −1

dmc ∝ mc−1/3 . τdouble = mc dt

(155)

(156)

In this growth regime, the more massive an embryo is, the faster it doubles its mass.

This mode of growth is called ‘runaway growth’, because planetesimals which are slightly more massive than their surrounding neighbours can grow very rapidly and detach themselves from the background mass distribution. This is how a swarm of planetesimals of initially similar masses can evolve into a system consisting of planetary embryos embedded in a planetesimal swarm.

We now consider the growth time scale in the runaway growth regime in more detail. We can write nm = ρs , the density of solids, in the disc and v ≃ Hs Ω, where Hs is the thickness of the solid layer. Equation (154) may now be written as 4/3

πGΣs mc dmc = dt v2

3 4πρc

1/3

Ω ,

(157)

where we have made the approximation ρs = Σs /(2Hs ), where Σs = surface density of solids, and have replaced the embryo radius Rc in terms of its mass, mc and density ρc . Integrating equation (157) gives 1/3

h i 3 πGΣs Ω −1/3 −1/3 t 3 mc (0) − mc (t) = v2 4πρc 50

(158)

which shows that the time for the embryo to grow from an initial mass mc (0) to a larger mass at time t, denoted by mc (t), depends inversely on the initial embryo mass.

The time required to grow to masses much larger than mc (0) may be approximated from equation (158) to be −1/3

3mc (0) t ≃ πGΣs

4πρc 3

1/3

Hs2 Ω

(159)

where we have used v = Hs Ω. If we take the values ρc ∼ 1 gm cm−3 (103 kg m−3 ) and Σs ∼ 1 gm cm−2 (10 kg m−2 ), (i.e. gas to dust ratio = 100), then we can write t ∼ 10

10

mc mEarth

−1/3

Hs R

2

M∗ M⊙

1/2

R R⊙

1/2

years.

(160)

Suppose that we have 10 km sized planetesimals with mc ≃ 1018 g = 1015 kg, and further assume that the dispersion velocity of the planetesimals is equal to the escape velocity from their surface (i.e. the planetesimal velocities are stirred by mutual gravitational interactions). In this case we find that Hs /R ∼ 10−4 , and the growth time of a planetary embryo of similar mass in orbit around a solar mass star at 1 AU is t ∼ 2 × 105 yr.

This number is significantly shorter than the expected life times of protostellar discs, as required.

7.5

The isolation mass

How large can a planetary embryo grow by accreting from the surrounding planetesimal swarm ? This question is answered by consideration of the feeding zone of an embryo, defined to be the surrounding annulus in the disc from which it is able to accrete planetesimals. The size of this annulus is set by the maximum distance over which the planetary embryo’s gravity is able to perturb planetesimal orbits so that they can collide. We define the half-width of this annulus by ∆amax = C.RHill where C is a constant and RHill is the embryo’s Hill radius defined by RHill = a

mc 3M∗

1/3

.

The isolation mass is given by miso = 2πa.2∆amax .Σs

(161)

where Σs is the surface density of solids (planetesimals), a is the semimajor axis of the embryo. Substituting for ∆amax gives miso = 4πa.C.a

51

miso 3M∗

1/3

.Σs ,

(162)

and solving for miso gives miso = 8π

3/2 3

aC

3/2

1 3M∗

1/2

Σ3/2 s .

(163)

Detailed of planetesimal orbits perturbed by a planet gives √ computation of the stability −2 C = 2 3. Taking Σs = 100 kg m at 1 AU gives miso = 0.07 M⊕ , suggesting that the terrestrial planets in our Solar System did not form by embryos simply accreting planetesimals in their feeding zones. At a distance of 5 AU, using the same solids surface density Σs = 100 kg m−1 , we obtain miso ≃ 9 M⊕ . This is in reasonable agreement with the estimated masses of the rock and ice cores at the centres of Saturn, Uranus and Neptune. The situation with Jupiter is more uncertain since here the core mass estimates are more sensitive to assumptions about the equation of state for hydrogen under mega-bar pressures – which is not well-constrained.

7.6

Overview of terrestrial planet formation

We conclude with a short discussion about terrestrial planet formation by summarizing briefly the main stages of the process: 1. Dust particles agglomerate to form, eventually, planetesimals. Most likely this occurs via pairwise collisions, though how (or whether) this process can work for cm to meter scale particles remains somewhat murky. There may be a role for gravitational instability.

2. Growth beyond planetesimals occurs via direct collisions, with an increasing role for gravitational focusing as masses become larger. Dynamical friction keeps the velocity dispersion of the most massive bodies low. Gas drag keeps the velocity dispersion of the the planetesimals low. A phase of runaway growth occurs in which a few bodies grow rapidly at the expense of the rest.

3. Runaway growth ceases once the largest bodies become massive enough to stir up the planetesimals in their vicinity. A phase of oligarchic growth ensues, in which the largest objects grow more slowly than they did during runaway growth, but still more rapidly than small bodies. Growth continues in this mode until the isolation mass is approached, at which point growth slows further.

4. Further evolution occurs as a result of collisions between the initially relatively isolated planetary embryos left over after oligarchic growth. The embryos are perturbed onto crossing orbits due to the influence of the giant planets and mutual secular resonances. The final assembly of the terrestrial planets takes around 100 Myr, with the predicted configuration varying depending upon the assumed surface density of planetesimals and existence (or not) of giant planets.

In the Solar System, the final giant impact on the Earth is widely considered to have given rise to the ejection of enough mass into orbit around the proto-Earth to subsequently form the Moon. This lunar-forming impact has been dated to have arisen at ∼ 60 Myrs after the formation of the Sun. This dating has been achieved using the isotopic ratios of 186 W and 182 W (Tungsten). Early in the 52

history of the Solar system, radioactive Hafnium 182 Hf decays into 182 W with a half-life of 9 Myrs. 182 Hf is considered to be a ‘lithophile’ element, which means that if the Earth becomes completely molten it will remain in the Silicate mantle and crust which forms on cooling. 182 W on the other hand is considered to be a ‘siderophile’ element, meaning that it prefers to follow the heavy element iron as it sinks to the centre of the Earth to form the core. By measuring the amout of 182 W to 186 W in the Earth’s crust at the present time, it is possible to calculate the time after the formation of the Sun when the Earth was last molten - which occured during the last lunar-forming giant impact.

The dominant uncertainties in theoretical models for terrestrial planet formation are arguably found during stage 1 — the formation of planetesimals. It is also true that most simulations of the late stages of terrestrial planet formation lead to planetary eccentricities that are slightly larger than those observed in the Solar System. This signals the need for additional dissipation at late epochs, possibly from the dynamical effect of a surviving population of smaller bodies.

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